Question: Determine how many solutions exist for the system of equations. ${-2x+y = 3}$ ${-3x+y = -4}$
Solution: Convert both equations to slope-intercept form: ${-2x+y = 3}$ $-2x{+2x} + y = 3{+2x}$ $y = 3+2x$ ${y = 2x+3}$ ${-3x+y = -4}$ $-3x{+3x} + y = -4{+3x}$ $y = -4+3x$ ${y = 3x-4}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 2x+3}$ ${y = 3x-4}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.